Exact Solutions and Scalar Fields in Gravity - Instituto Avanzado de ...

Quantum cosmology with self–interacting scalar field 201
3.1. EXACT SOLUTIONS FOR BIANCHI I
WITH
For Bianchi type I cosmological model We restrict ourselves
to the case i.e. this potential
has been used in classical Bianchi type I vacuum cosmology [10] and for
isotropic models with a barotropic fluid[11]; this potential is one of those
that could produce inflation[4]. This choice simplifies the WDW equation
and allows us to obtain exact solutions,
where and the are constants.
The following wavefunction can be obtained by superposition or it can
be substituted into the WDW equation to check that it is a particular
solution
where r is an arbitrary real number; satisfy the relations
For a
wormhole Taking the negative root for we can check that
the wavefunction is exponentially damped for large spatial geometry,
i.e., when and also that it does not oscillate when
3.2. SOLUTIONS FOR BIANCHI I WITH
ANOTHER POTENTIAL
We take another potential for which it is possible to obtain exact
solutions to WDW equation. In Eq.(26) if we take the
action becomes,
We change variables and take a potential,
The corresponding WDW equation is given by